# Conjugate

If a polytope is realized such that its vertices lie at coordinates in a field, a **conjugate** polytope is formed by transforming the vertices using an automorphism of the field. The resulting polytope is combinatorially identical. Although commonly viewed as an operation, it is slightly more accurately viewed as an equivalence relation; a polytope may have multiple conjugates, or may not have any conjugates other than itself.

The field in question is usually chosen to be the field of algebraic numbers **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mathbb{Q}}}**
, whose realization admits many common polytopes defined by symmetry. For example, many convex regular polygons have conjugates in the form of regular star polygons; in particular, the regular heptagon, the heptagram, and the great heptagram are all conjugates. The conjugates of regular-faced polyhedra are the results of symmetrically replacing faces with conjugates; for example, the small stellated dodecahedron and great dodecahedron are conjugates.

The conjugates of uniform polytopes are always uniform, and the conjugates of scaliform polytopes are always scaliform. Because of this, conjugates can be a useful tool to find new polytopes with such properties.

## Definition[edit | edit source]

A **conjugate** of a polytope **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P}**
with coordinates in a field **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F }**
is the polytope **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{P}}**
created when an automorphism of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F }**
is applied to the coordinates of the vertices of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P}**
. The number of conjugates of a polytope depends on the number of automorphisms of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F}**
(see Choice of field).

## Algebraic conjugates[edit | edit source]

To understand the effect conjugation has on the coordinates of a polytope, we need to look into a related mathematical notion.

An **algebraic number** is any real number **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi}**
that's the root of some polynomial with rational coefficients. That is to say, there exist rational **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0,\ldots,a_n}**
such that

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0+a_1\xi+a_2\xi^2+\ldots+a_n\xi^n=0.}**

Every algebraic number has an associated **minimal polynomial**, which is the monic polynomial (its leading coefficient is 1) of least degree of which it is a root. Two algebraic numbers are said to be (algebraically) **conjugate** if they have the same minimal polynomial. Being conjugate is an equivalence relation.

It turns out that automorphisms of fields of real numbers must necessarily send numbers to algebraic conjugates, as they must satisfy the same algebraic equations involving rational numbers. In particular, this implies that the coordinates of conjugate polytopes are always algebraic conjugates of one another, hence the name. Note however that arbitrarily swapping coordinates by their algebraic conjugates won't necessarily yield a conjugate.

## Properties[edit | edit source]

Conjugates do not depend on the position, size, or orientation of the original polytope, as long as these are changed within the same field. The resulting conjugate may also be transformed, but its shape does not change. A brief proof follows:

- Translating a polytope by a vector
**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}}**translates the conjugate polytope by its component-wise conjugate**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\mathbf{v}}}**, since field automorphisms respect addition. - Likewise, scaling a polytope by a factor
**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s}**scales the conjugate polytope by its conjugate**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{s}}**, since field automorphisms respect multiplication. - Rotating and/or reflecting a polytope is equivalent to multiplying its coordinates by a matrix
**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q}**such that**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q^TQ=I}**. The coordinates of the conjugate polytope, then, are multiplied by its element-wise conjugate**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Q}}**. Since field automorphisms respect matrix multiplication,**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Q}^T\overline{Q} = \overline{I}}**, and since automorphisms also preserve the identity matrix,**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Q}^T\overline{Q} = I}**. Thus**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Q}}**also represents a rotation and/or reflection of the conjugate polytope.

Furthermore, conjugates of planar polytopes will also be planar, as field automorphisms respect bases for subspaces. Additionally, conjugate polytopes always have the same symmetries, and the same amounts of elements in each dimension. Some corresponding elements may not be exactly the same, but will be conjugates of each other.

## Choice of field[edit | edit source]

Polytopes are often defined with real coordinates, but **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}}**
has only the trivial automorphism,^{[1]} which does not allow for non-trivial conjugates. Instead, the coordinate field is restricted to an algebraic field, e.g. **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}\left[\sqrt{5}\right]}**
, which may have non-trivial automorphisms.

It appears that many polytopes **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P}**
have a "canonical field" **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F}**
, which is just large enough to represent the coordinates of some position, size, and orientation of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F=\mathbb{Q}\left[\sqrt{5}\right]}**
, but it cannot be represented in **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^2}**
and must be embedded in **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^3}**
instead (i.e. as a face of the dodecahedron). Additionally, not all automorphisms of the canonical field may create real conjugate polytopes, as they may output complex numbers.

## Examples[edit | edit source]

- The rational numbers have no automorphisms besides the identity, so all polytopes which may be written with rational coordinates, such as the octahedron and icositetrachoron, have no non-trivial conjugates.
- Regular polygons with the same number of sides and connected components (e.g. the heptagon, heptagram, and great heptagram) are all conjugates.
- Snid, gosid, gisid, and girsid are conjugate polyhedra whose coordinates lie in a sextic field, four of whose automorphisms preserve real numbers.
- The conjugates of a prism product of polytopes
**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q}**whose canonical fields share no automorphisms are the prism products of the conjugate(s) of**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q}**. For example, the 5-8, 5-8/3, 5/2-8, and 5/2-8/3 duoprisms are conjugates. However, if

## References[edit | edit source]

- ↑ Makoto Kato (2013). "Is an automorphism of the field of real numbers the identity map?".

## External links[edit | edit source]

- Wikipedia contributors. "Algebraic number".
- Wikipedia contributors. "Conjugate element (field theory)".